Bed bugs

I think we have bed bugs. Over the last week or so I’ve been coming out in little red spots all over my body (not many of them, just widespread) and didn’t really pay much attention to them until I noticed one on my sons arm. Not 100% it’s bed bugs but I checked my bed and found what looked like a leg of a bug. I washed and dried all bedding on a high heat, hoovered our rooms, bed frames, mattresses, etc, and then a few days later I found in my bed a small light brown shell type thing, like a sesame seed but translucent.

Does this sound like bed bugs? No idea where they’ve come from if it is but desperate to get rid. What’s the best way of doing so?

Thanks!